Gravitational and Elastic Potential Energy — AP Physics 1
1. Core Concepts of Potential Energy ★★☆☆☆ ⏱ 3 min
Potential energy is stored energy in a closed system arising from the relative positions of interacting objects. Gravitational potential energy is stored due to gravitational interaction between masses; for AP Physics 1, we only work with values near Earth's surface where gravity is approximately constant. Elastic potential energy is stored when an elastic object like an ideal spring is stretched or compressed from equilibrium. By convention, we use $U$ for potential energy, with $U_g$ for gravitational and $U_s$ (or $U_{el}$) for elastic. Only changes in potential energy are physically meaningful, so we can choose any reference point to set $U=0$. This topic makes up ~6-9% of your total AP Physics 1 exam score, appearing in both MCQ and FRQ.
2. Gravitational Potential Energy Near Earth's Surface ★★☆☆☆ ⏱ 4 min
Gravitational potential energy describes stored energy from separation between masses in a gravitational field. Near Earth's surface, the gravitational force on a mass $m$ is approximately constant, pointing downward. The change in gravitational potential energy of the mass-Earth system equals the negative of the work done by gravity when the mass changes position:
\Delta U_g = -W_g
If the mass changes height by $\Delta y = y_{final} - y_{initial}$ (with upward as positive), gravity does work $W_g = -mg\Delta y$. Substituting gives:
\Delta U_g = mg\Delta y
If we choose a reference point where $U_g = 0$, we can write $U_g = mgy$, where $y$ is the height of the mass relative to that reference. Increasing the height of the mass increases stored gravitational potential energy, which makes intuitive sense, as the stored energy can be converted to motion when the object falls.
Exam tip: Always confirm that $\Delta y$ is the change in vertical height, not horizontal distance or trail length—AP 1 questions often give trail length to test this distinction.
3. Elastic Potential Energy for Ideal Springs ★★★☆☆ ⏱ 4 min
Elastic potential energy is stored in an ideal spring (or other elastic material) when it is stretched or compressed from its equilibrium (relaxed) position. An ideal spring obeys Hooke's Law:
F_s = -kx
where $x$ is displacement from equilibrium, and $k$ is the spring constant (a measure of stiffness, units N/m). To find elastic potential energy, we use the relation that change in potential energy equals negative work done by the conservative spring force. The work done by the spring moving from equilibrium ($x=0$) to displacement $x$ is $W_s = -\frac{1}{2}kx^2$, so we get the standard formula, with $U_s = 0$ at equilibrium (the standard AP 1 convention):
U_s = \frac{1}{2}kx^2
Notice that $x$ is squared, so stretching a spring by 0.1 m stores the same amount of energy as compressing it by 0.1 m. Stiffer springs (higher $k$) store more energy for the same displacement, and energy grows with the square of displacement, so doubling displacement quadruples stored energy.
Exam tip: Never use the total length of the spring for $x$—$x$ is always displacement from equilibrium (relaxed length), so you must subtract the relaxed length from the stretched/compressed length to get $x$.
4. Conservation of Mechanical Energy with Multiple Potential Energies ★★★★☆ ⏱ 7 min
Total mechanical energy of a closed system is the sum of kinetic energy, gravitational potential energy, and elastic potential energy:
E_{mech} = K + U_g + U_s
If no non-conservative forces (friction, air resistance, external pushes) do net work on the system, total mechanical energy is conserved. This means initial total mechanical energy equals final total mechanical energy:
K_i + U_{g,i} + U_{s,i} = K_f + U_{g,f} + U_{s,f}
This is one of the most useful problem-solving tools in AP Physics 1, because it lets you find speed or position without calculating acceleration or forces at every point along the motion. To simplify calculations, always choose a convenient zero reference for $U_g$, usually setting $U_g = 0$ at the lowest point of motion, so that term becomes zero in the final calculation.
Exam tip: Always sketch initial and final positions to confirm all displacement/height changes—students almost always forget to add the spring compression to the total height change in this type of problem.
Common Pitfalls
Why: Questions often give both relaxed and stretched length to test understanding, and students default to the larger number given.
Why: Students stop at the height of the object above the relaxed spring, and ignore the additional distance the object falls while compressing the spring.
Why: Students assume potential energy can never be negative, forgetting only changes in potential energy are physically meaningful.
Why: Students confuse the linear force-displacement Hooke's Law relation with the quadratic energy-displacement relation.
Why: Textbooks often use shorthand like "the hiker's potential energy," leading to this mistake, which AP 1 explicitly tests.